I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$.
A linked question could be: does there exist a Whitehead double with genus $1$?
Thanks to all!
Both the questions have affirmative answers. Moreover, it is true that the Whitehead double of every non-trivial knot is a knot with genus $1$.
In fact, if you consider the pattern $K'$ contained in the solid torus $D^2 \times S^1$, it is quite easy to construct a Seifert surface $S$ for $K'$ with genus $1$ and contained in the solid torus. Such a surface is built with two rings and one of them is twisted.
Added (Jim Conant): Here are a couple of pictures of the genus one Seifert surface for $Wh(K)$.